Symmetries and Groups in Quantum Mechanics

This lecture focuses on the foundational concepts of symmetries and groups in quantum mechanics. The instructor begins by discussing the importance of symmetries, emphasizing that they are realized through representations in a specific space. The lecture covers various groups, particularly SU2 and SU3, and their properties, including simply connectedness and the implications for quantum mechanics. The instructor illustrates how SU2 serves as the universal cover of its algebra, explaining the relationship between the algebraic structure and the topology of the group. The discussion extends to the role of projective representations and the significance of the Lorentz group in the context of quantum mechanics. The instructor highlights the differences between connected and simply connected groups, using examples to clarify these concepts. The lecture concludes with a brief overview of the Poincaré algebra and its relevance to the study of symmetries in physics, setting the stage for future discussions on fits and further applications of these principles in modern physics.